PASSIVE CONTROL OF VIBRATION AND WAVE PROPAGATION IN SANDWICH PLATES WITH PERIODIC AUXETIC CORE

1. PASSIVE CONTROL OF VIBRATION AND WAVE PROPAGATION IN SANDWICH PLATES WITH PERIODIC AUXETIC CORE Luca Mazzarella Mechanical Engineering Dept. Catholic University of America Washington, DC 20064 Massimo Ruzzene Mechanical Engineering Dept. Catholic University of America Washington, DC 20064 Panagiotis Tsopelas Civil Engineering Dept. Catholic University of America Washington, DC 20064

2. Outline • Introduction: wave propagation in 1D and 2D periodic structures; • Sandwich plates with periodic honeycomb core: • Finite Element Modeling of unit cell • Bloch reduction • Response to harmonic loading • Performance of periodic sandwich plates: • Configuration of the unit cell • Phase constant surfaces • Contour plots • Harmonic response • Sandwich plate-rows with periodic honeycomb core: • Theoretical Modeling • Transfer Matrix • Propagation constants • Dynamic stiffness matrix • Performance of periodic sandwich plate-rows: • Configuration of the unit cell • Propagation Patterns • Structural response • Conclusions

3. y Core B Core A x Motivation Analysis of WAVE DYNAMICS in sandwich plates with core of two honeycomb materials alternating PERIODICALLY along the structure • Analysis is performed through the theory of 2-D PERIODIC STRUCTURES which are characterized by: • Frequency bands where elastic waves do not propagate • Directions where propagation of elastic waves does not occur STOP/PASS BANDS “FORBIDDEN” ZONES

4. Motivation 2D periodic structures behave as DIRECTIONAL MECHANICAL FILTERS • GOAL:Evaluate characteristics of wave propagation for sandwich plates with periodic core configuration: • Determine stop/pass band pattern; • Determine directional characteristic and “forbidden zones” of response; • Evaluate influence of the cell and core geometry;

5. Completely passive treatment Performance of traditional light-weight sandwich elements enhanced by directional filtering capabilities Improvement of the attenuation capabilities of periodic sandwich panels obtained through a proper selection of the core and cell configuration Stiffening geometric effect and change in mass density depending on core material geometry External dimensions and weight not significantly affected Basic concepts Sandwich plates with periodic auxetic core:

6. U U U U L R L R Cell k-1 Cell k k-1 k k k-1 F F L R k k Interface k F F R L k-1 k-1 Introduction: plate-rows Periodic structure:assembly of identical elementary components, or cells, connected to one another in a regular pattern. The plate-rows here considered are modeled as quasi-one-dimensional multi-coupled periodic systems Transfer matrix formulation: Yk : state vector Tk : transfer matrix i.e. • li  =1 : pass band (wave propagation) • li  1 : stop band (wave attenuation) li : ith eigenvalue of Transfer Matrix T log(li ) = PROPAGATION CONSTANT

7. Lx ny Ly y x Unit cell nx Motion of unit cell Propagation Constants Introduction: 2D plates 2D-periodic structure: cells connected to cover a plane Wave motion in the 2-D structure (Bloch’s Theorem):

8. Introduction: 2D plates Propagation Constants are complex numbers: (k=x,y) Phase Constant Attenuation Constant Condition for wave propagation: & Imposing the propagation constants allows obtaining the corresponding frequency of wave propagation: Phase Constant Surfaces

9. Face sheets Honeycomb core A Honeycomb core B Honeycomb core Geometric layout of regular (A) and auxetic (B) honeycomb structures q< 0: Re-entrant geometry (AUXETIC SOLID) t t l l q q a=h/l b=t/l h h (core B) (core A) Negative Poisson’s ratio behavior: Auxetic honeycombs with q=-60°, a=2 are characterized by a shear modulus which outcast up to five times the shear modulus of a regular honeycomb of the same material

10. Core B x Simply supported edges Core A y Wave propagation in sandwich plate-rows Plate-Rows Outline of concepts described and methods applied • SFEM is formulated from Transfer Matrix approach • Transfer Matrix obtained from distributed parameter model of sandwich plate • Transfer Matrix is recast to obtain the Dynamic Stiffness Matrix of plate element TRANSFER MATRIX PASS / STOP BANDS ASSEMBLED DYNAMIC STIFFNESS MATRIX RESPONSE OF STRUCTURE

11. Equations of motion Generalized Displacements Generalized Forces Transfer Matrix (kth part of the cell) Transfer Matrix formulation • Only the dynamics along the x-axis has to be investigated • The behavior along the y-axis is described by the harmonic exp(jkyy); • ky=mp/Ly (with m integer) is the wave number along the y-axis • The analysis is performed independently for each harmonic m for the deformations along the y-axis: state space formulation: according to the CLT (Classical Laminated plate Theory).

12. Performance of periodic sandwich plate-rows Cell configuration and Geometry: Honeycomb core A material thickness Face sheet (1) Al 5 mm core (2) Nomex 9 cm Face sheet (3) Al 5 mm Simply supported edges Face sheets Auxetic core B • Aspect ratio Lx/ Ly=1/2 • Length ratio LxA/ LxB=1, 2, 1/2 • Internal core geometry: Type A: a=1, q=30 Type B: a =2, q=-60 and q=-30 • Ly=1 m LxA Ly LxB Output Unit cell Core “A” LxA Simply supported edges Core “B” LxB Configuration of periodic plate-row: • 20 cells along S-S edges Forcing function F=F0sin(mpy/Ly)ejwt x Ly y

13. ky=p/Ly Single cell eigenvalues 20 cells plate-row FRF LxA/LxB=1; qB=-60º LxA/LxB=1/2; qB=-60º Performance of periodic sandwich plate-rows

14. Single cell eigenvalues 20 cells plate-row FRF LxA/LxB=1; qB=-60º Performance of periodic sandwich plate-rows ky=2p/Ly LxA/LxB=2; qB=-60º

15. Single cell eigenvalues 20 cells plate-row FRF LxA/LxB=1; qB=-60º LxA/LxB=1/2; qB=-60º Performance of periodic sandwich plate-rows ky=3p/Ly

16. Performance of periodic sandwich plate-rows Propagation patterns ky=mp/Ly ; m=1,…,10 10 10 9 9 8 8 7 7 Forcing harmonic Forcing harmonic Homogeneous core A 6 6 LxA/LxB=1; qB=-30º 5 5 4 4 3 3 2 2 1 1 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 [Hz] [Hz] [Hz] 10 10 9 9 8 8 7 7 Forcing harmonic Forcing harmonic 6 6 LxA/LxB=1/2; qB=-30º LxA/LxB=2; qB=-60º 5 5 4 4 3 3 2 2 1 1 0 50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 [Hz] [Hz]

17. {qLT},{FLT} {qRT},{FRT} {qT},{FT} Top Left {qI},{FI} {qL},{FL} {qR},{FR} Internal Right Bottom {qLB},{FLB} {qRB},{FRB} {qB},{FB} [K], [M]: cell`s stiffness and mass matrices 2D analysis: Bloch reduction Cell analysis: {qij} generalized displacements at interface i,j {Fij} generalized forces at interface i,j Cell`s equation of motion: where:

18. 2D analysis: Bloch reduction • Bloch`s Theorem: • Relation between interface displacements (compatibility conditions) • Relation between interface forces (equilibrium conditions) • Reduced Mass and Stiffness Matrices: with: • Cell`s Equation of Motion is reduced at: Frequency w of wave motion for the assigned set of propagation constants mx, my

19. w (rad/s) 1 1 ey/p ex/p 0 2D Wave propagation Solution of Dispersion Relation: Phase Constant Surfaces: w=w(ex, ey ) • Phase Constant Surfaces are symmetric with respect to both ex, ey • Analysis can be limited to the first quadrant of the ex, ey plane, within • the [0,p] range for ex, ey . First three phase constant surfaces for a sandwich plate with uniform core (A) represented over the first propagation zone ([0,p] range for ex, ey )

20. Honeycomb core A Face sheets Auxetic core B LyA Lx LyB Performance of periodic sandwich plates Cell configuration and Geometry: material thickness [mm] Face sheet (1) Al 1 core (2) Nomex 2 Face sheet (3) Al 1 • Aspect ratio Lx/ Ly_=1 • Length ratio LyA/ LyB_=1 and 2/3 • Internal core geometry: Type A: a=1, q=30 Type B: a =2, q=-60 Unit cell Harmonic forcing Core “A” LyA Core “B” LyB Configuration of periodic plate: Lx

21. 14 15 12 13 11 9 10 7 8 5 6 4 3 2 ey/p 1 0.9 w (rad/s) 0.8 0.7 0.6 0.5 ex/p 1 0.4 1 ey/p 0.3 ex/p 0 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Phase Constant Surfaces • First phase constant surface • Contour plot Homogeneous core, Lx/ Ly_=1 The energy flow vector P at a given frequency w lies along the normal to the corresponding iso-frequency contour line in the kx, ky space, where ki=ei/Li , (i=x,y). The perpendicular to a given iso-frequency line for an assigned pair ex, ey corresponds to the direction of wave propagation(*) (*) Langley R.S., “The response of two dimensional periodic structures to point harmonic forcing” JSV (1996)197(4), 447-469.

22. 14 20 15 18 19 12 16 17 13 11 14 11 15 12 9 8 10 7 13 8 6 5 10 6 9 4 7 4 3 5 2 3 2 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Contour plots Influence of the periodic core: Homogeneous core, Lx/ Ly_=1 Periodic core, Lx/ Ly_=1 , LyA/LyB=1 ey/p ey/p ex/p ex/p • Periodic core (length ratio =1): • Directional behavior expected • above a “transition frequency” • Homogeneous core: • No directional behavior expected

23. Plate harmonic response Plate deformed configuration for excitation at w=7.5 rad/s Homogeneous LyA/LyB=1 LyA/LyB=2/3 Number of cells: Nx= 20, Ny= 20; 40x40 finite element grid

24. Plate harmonic response Plate deformed configuration for excitation at w=9.5 rad/s Homogeneous LyA/LyB=1 LyA/LyB=2/3 Number of cells: Nx= 20, Ny= 20; 40x40 finite element grid

25. Plate harmonic response Plate deformed configuration for excitation at w=13 rad/s Homogeneous LyA/LyB=1 LyA/LyB=2/3 Number of cells: Nx= 20, Ny= 20; 40x40 finite element grid

26. Conclusions • Wave propagation in periodic sandwich plates and plate-rows is analyzed; • Auxetic and regular honeycomb cellular solids are utilized as core materials to generate impedance mismatch zones; • Analysis is performed through the combined application of the theory of periodic structures, the FE method, and the Transfer matrix and Spectral FE methods • The capability of the periodic core to generate stop bands for the propagation of waves along the plate-rows, and directional patterns for the propagation of waves along the plate plane has been assessed; • Analysis allows evaluating pass/stop bands propagation patterns, and the phase constant surfaces for the estimation of directional characteristics for wave propagation • The filtering capabilities are influenced by the geometry of the periodic cell; • Harmonic response shows directionality at specified frequencies, and confirms the propagation patterns • Completely passive treatment; • External dimensions and weight not significantly affected; • Improvement of the attenuation capabilities of periodic sandwich panels obtained through a proper selection of the core and cell configuration;